5.01a Permutations and combinations: evaluate probabilities

1. 1. A security code is made up of 4 numerical digits followed by 3 distinct uppercase letters.

Given that the digits must be from the set \(\{ 1,2,3,4,5 \}\) and the letters from the set \{A, B, C, D\}

1. determine the total number of possible codes using this system. To enable more codes to be generated, the system is adapted so that the 3 letters can appear anywhere in the code but no letter can be next to another letter.
2. Determine the increase in the number of codes using this adapted system.
   (ii) A combination lock code consists of four distinct digits that can be read as a positive integer, \(N = a b c d\), satisfying
   • all the digits are odd
   • \(\quad N\) is divisible by 9
   • the digits appear in either ascending or descending order
   • \(\quad N \equiv e ( \bmod a b )\) where \(a b\) is read as a two-digit number and \(e\) is the odd digit that is not used in the code
   • Use the first two properties to determine the four digits used in the code.
   • Hence determine the code on the lock.

1. (i) Use the Euclidean algorithm to find the highest common factor of 602 and 161.

Show each step of the algorithm.
(ii) The digits which can be used in a security code are the numbers \(1,2,3,4,5,6,7,8\) and 9. Originally the code used consisted of two distinct odd digits, followed by three distinct even digits. To enable more codes to be generated, a new system is devised. This uses two distinct even digits, followed by any three other distinct digits. No digits are repeated. Find the increase in the number of possible codes which results from using the new system.

1. A random sample of 150 observations is taken from a geometric distribution with parameter 0.3

Estimate the probability that the mean of the sample is less than 3.45

1. There are 32 students in a class.

Each student rolls a fair die repeatedly, stopping when their total number of sixes is 4 Each student records the total number of times they rolled the die. Estimate the probability that the mean number of rolls for the class is less than 27.2

1. A random sample of 100 observations is taken from a Poisson distribution with mean 2.3

Estimate the probability that the mean of the sample is greater than 2.5

6 Answer parts (ii)(A) and (iii)(B) of this question on the insert provided. A particular component of a machine sometimes fails. The probability of failure depends on the age of the component, as shown in Table 6. \section*{Table 6} You are to simulate six years of machine operation to estimate the probability of the component failing during that time. This will involve you using six 2-digit random numbers, one for each year.

1. Give a rule for using a 2-digit random number to simulate failure of the component in its first year of life. Similarly give rules for simulating failure during each of years 2 to 6 .
2. (A) Use your rules, together with the random numbers given in the insert, to complete the simulation table in the insert. This simulates 10 repetitions of six years operation of the machine. Start in the first column working down cell-by-cell. In each cell enter a tick if there is no simulated failure and a cross if there is a simulated failure. Stop and move on to the next column if a failure occurs.
   (B) Use your results to estimate the probability of a failure occurring. It is suggested that any component that has not failed during the first three years of its life should automatically be replaced.
3. (A) Describe how to simulate the operation of this policy.
   (B) Use the table in the insert to simulate 10 repetitions of the application of this policy. Re-use the same random numbers that are given in the insert.
   (C) Use your results to estimate the probability of a failure occurring.
4. How might the reliability of your estimates in parts (ii) and (iii) be improved?

1 Bill and Gill send letters to potential sponsors of a show. On past experience, they know that \(5 \%\) of letters receive a favourable reply.

1. Bill sends a letter to each of 40 potential sponsors. Assuming that the number \(N\) of favourable responses can be modelled by a binomial distribution, find the mean and variance of \(N\).
2. Gill sends one letter at a time to potential sponsors. \(L\) is the number of letters she sends, up to and including the first letter that receives a favourable response.
   1. State two assumptions needed for \(L\) to be well modelled by a geometric distribution.
   2. Using the assumptions in part (ii)(a), find the smallest number of letters that Gill has to send in order to have at least a \(90 \%\) chance of receiving at least one favourable reply.

2 Each letter of the words NEW COURSE is written on a card (including one blank card, representing the space between the words), so that there are 10 cards altogether.

1. All 10 cards are arranged in a random order in a straight line. Find the probability that the two cards containing an E are next to each other.
2. 4 cards are chosen at random. Find the probability that at least three consonants ( \(\mathrm { N } , \mathrm { W } , \mathrm { C } , \mathrm { R } , \mathrm { S }\) ) are on the cards chosen.

3 Adila has a pack of 50 cards.

1. Each of the 50 cards is numbered with a different integer from 1 to 50 . Adila selects 5 cards at random without replacement.
   1. Find the probability that exactly 3 of the 5 cards have numbers which are 10 or less.
   2. Adila arranges the 5 cards in a line in a random order. Find the probability that the 5 cards are arranged in numerically increasing order. 10 of the 50 cards are blue and the rest are green.
   3. Adila randomly selects three sets of 10 cards each, without replacement. The sets are labelled \(A , B\) and \(C\). Given that \(A\) contains 3 blue cards and 7 green cards, find the probability that \(B\) contains exactly 2 blue cards and \(C\) contains exactly 3 blue cards.

3 A pack of 40 cards consists of 10 cards in each of four colours: red, yellow, blue and green. The pack is dealt at random into four "hands", each of 10 cards. The hands are labelled North, South, East and West.

1. Find the probability that West has exactly 3 red cards.
2. Find the probability that West has exactly 3 red cards, given that East and West have between them a total of exactly 5 red cards.
3. South has 5 red cards and 5 blue cards. These cards are placed in a row in a random order. Find the probability that the colour of each card is different from the colour of the preceding card.

1

1. (a) Show that the number of arrangements of 25 distinct objects is an integer multiple of \(5 ^ { 6 }\).
   (b) Explain how this shows that the number of arrangements of 25 distinct objects is a whole number of millions.
2. (a) Calculate the values of

5

1. How many arcs does the complete bipartite graph \(K _ { 5,5 }\) have? A subgraph of \(K _ { 5,5 }\) contains 5 arcs joining each of the elements of the set \(\{ 1,2,3,4,5 \}\) to an element in a permutation of the set \(\{ 1,2,3,4,5 \}\). Suppose that \(r\) is connected to \(p ( r )\) for \(r = 1,2,3,4,5\).
2. How many permutations would have \(p ( 1 ) \neq 1\) ?
3. Using the pigeonhole principle, show that for every permutation of \(\{ 1,2,3,4,5 \}\), the product \(\Pi _ { r = 1 } ^ { 5 } ( r - p ( r ) )\) is even (i.e. an integer multiple of 2, including 0 ).
4. Is the result in part (iii) true when the permutation is of the set \(\{ 1,2,3,4,5,6 \}\) ? Give a reason for your answer.

3 A discrete random variable \(X\) has the distribution \(\mathrm { U } ( 11 )\).
The mean of 50 observations of \(X\) is denoted by \(\bar { X }\).
Use an approximate method, which should be justified, to find \(\mathrm { P } ( \bar { X } \leqslant 6.10 )\).

6 A bag contains 7 red counters and 5 blue counters.

1. Fred chooses 4 counters at random, without replacement. Show that the probability that Fred chooses exactly 2 red counters is \(\frac { 14 } { 33 }\).
2. Lina chooses 4 counters at random from the bag, records whether or not exactly 2 red counters are chosen, and returns the counters to the bag. She carries out this experiment 99 times.
   1. Find the mean of the number of experiments that result in choosing exactly 2 red counters.
   2. Find the variance of the number of experiments that result in choosing exactly 2 red counters.
   3. Alex arranges all 12 counters in a random order in a straight line. A is the event: no two blue counters are next to one another. B is the event: all the blue counters are next to one another. Find \(\mathrm { P } ( A \cup B )\).

3

1. Alex places 20 black counters and 8 white counters into a bag. She removes 8 counters at random without replacement. Find the probability that the bag now contains exactly 5 white counters.
2. Bill arranges 8 blue counters and 4 green counters in a random order in a straight line. Find the probability that exactly three of the green counters are next to one another.

3 A set of ten cards have the following values: \(\begin{array} { l l l l l l l l l l } 13 & 8 & 4 & 20 & 12 & 15 & 3 & 2 & 10 & 8 \end{array}\) Kerenza wonders if there is a set of these cards with a total of exactly 50 .

1. Which type of problem (existence, construction, enumeration or optimisation) is this? The five cards \(4,8,8,10\) and 20 have a total of 50.
2. How many ways are there to arrange three of these five cards (with the two 8 s being indistinguishable) so that the total of the numbers on the first two cards is less than the number on the third card?
3. How many ways are there to select (choose) three of the five cards so that the total of the numbers on the three cards is less than 25 ?
4. Show how quicksort works by using it to sort the original list of ten cards into increasing order.
   You should indicate the pivots used and which values are already known to be in their correct position.

2 A group of 6 people is to be chosen from 4 men and 11 women.

1. In how many different ways can a group of 6 be chosen if it must contain exactly 1 man?
   Two of the 11 women are sisters Jane and Kate.
2. In how many different ways can a group of 6 be chosen if Jane and Kate cannot both be in the group?

4

1. In how many different ways can the 9 letters of the word TELESCOPE be arranged?
2. In how many different ways can the 9 letters of the word TELESCOPE be arranged so that there are exactly two letters between the T and the C ?

7. A bag contains \(n\) marbles of which 7 are green. From the bag, 3 marbles are selected at random.
The random variable \(X\) represents the number of green marbles selected.
The cumulative distribution function of \(X\) is given by

1. Show that \(n ( n - 1 ) ( n - 2 ) = 7980\)
2. Verify that \(n = 21\) satisfies the equation in part (a). Given that \(n = 21\)
3. find the exact value of \(a\) and the exact value of \(b\)
   \includegraphics[max width=\textwidth, alt={}]{fa1cb8a2-dab9-4133-b7a1-9108888c37d7-28_2655_1947_114_116}

5 In a random sample of 12 bags of flour, the weight, in grams, of flour in each bag was recorded as follows. \(\begin{array} { l l l l l l l l l l l l } 1011 & 995 & 1018 & 1022 & 1014 & 1005 & 1017 & 1015 & 993 & 1018 & 992 & 1020 \end{array}\)

1. It may be assumed that the weight of flour in a bag is normally distributed with a standard deviation of 10.5 grams.
   1. Construct a \(98 \%\) confidence interval for the mean weight, \(\mu\) grams, of flour in a bag, giving the limits to four significant figures.
   2. State why, in constructing your confidence interval, use of the Central Limit Theorem was not necessary.
   3. If the distribution of the weight of flour in a bag was unknown, indicate a minimum number of weights that you would consider necessary for a confidence interval for \(\mu\) to be valid.
2. The statement ' 1 kg ' is printed on each bag. Comment on this statement using both the confidence interval that you constructed in part (a)(i) and the weights of the given sample of 12 bags.
3. Given that \(\mu = 1000\), state the probability that a \(98 \%\) confidence interval for \(\mu\) will not contain 1000.
   (l mark)

6 On arrival at a business centre, all visitors are required to register at the reception desk. An analysis of the register, for a random sample of 100 days, results in the following information on the number, \(X\), of visitors per day.

1. Calculate an estimate of:
   1. \(\mu\), the mean number of visitors per day;
   2. \(\sigma\), the standard deviation of the number of visitors per day.
2. Give a reason, based upon the data provided, why \(X\) is unlikely to be normally distributed.
3. 1. Give a reason why \(\bar { X }\), the mean of a random sample of 100 observations on \(X\), may be assumed to be normally distributed.
   2. State, in terms of \(\mu\) and \(\sigma\), the mean and variance of \(\bar { X }\).
4. Hence construct a \(99 \%\) confidence interval for \(\mu\).
5. The receptionist claims that she registers on average more than 30 visitors per day, and frequently registers more than 50 visitors on any one day. Comment on each of these two claims.

2 The heights of sunflowers may be assumed to be normally distributed with a mean of 185 cm and a standard deviation of 10 cm .

1. Determine the probability that the height of a randomly selected sunflower:
   1. is less than 200 cm ;
   2. is more than 175 cm ;
   3. is between 175 cm and 200 cm .
2. Determine the probability that the mean height of a random sample of 4 sunflowers is more than 190 cm .

7

1. A greengrocer displays apples in trays. Each customer selects the apples he or she wishes to buy and puts them into a bag. Records show that the weight of such bags of apples may be modelled by a normal distribution with mean 1.16 kg and standard deviation 0.43 kg . Determine the probability that the mean weight of a random sample of 10 such bags of apples exceeds 1.25 kg .
2. The greengrocer also displays pears in trays. Each customer selects the pears he or she wishes to buy and puts them into a bag. A random sample of 40 such bags of pears had a mean weight of 0.86 kg and a standard deviation of 0.65 kg .
   1. Construct a \(\mathbf { 9 6 \% }\) confidence interval for the mean weight of a bag of pears.
   2. Hence comment on a claim that customers wish to buy, on average, a greater weight of apples than of pears.
      [0pt] [2 marks]